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Notes
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The setting of initial penalty error costs can affect the path of the optimization
and, indeed, whether a solution is achieved at all. If the penalty costs are too low, then
unbounded solutions may result although Xpress-SLP will increase the costs in an attempt to
recover. If the penalty costs are too high, then the requirement to achieve feasibility of
the linearized constraints may be too strong to allow the system to explore the nonlinear
feasible region. Low penalty costs can result in many SLP iterations, as feasibility of the
nonlinear constraints is not achieved until the penalty costs become high enough; high
penalty costs force feasibility of the linearizations, and so tend to find local optima close
to an initial feasible point.
Xpress-SLP can analyze the problem to estimate the size of penalty costs required to avoid
an initial unbounded solution. XSLP_OBJTOPENALTYCOST can be used in conjunction
with this procedure to scale the costs and give an appropriate initial value for balancing the
requirements of feasibility and optimality.
Not all models are amenable to the Xpress-SLP analysis. As the analysis is initially
concerned with establishing a cost level to avoid unboundedness, a model which is
sufficiently constrained will never show unboundedness regardless of the cost. Also, as the
analysis is done at the start of the optimization to establish a penalty cost, significant
changes in the coefficients, or a high degree of nonlinearity, may invalidate the initial
analysis.
A setting for XSLP_OBJTOPENALTYCOST of zero disables the analysis. A setting of 3 or 4 has proved successful for many models. If XSLP_OBJTOPENALTYCOST
cannot be used because of the problem structure, its effect can still be emulated by
some initial experiments to establish the cost required to avoid unboundedness, and then
manually applying a suitable factor. If the problem is initially unbounded, then the penalty cost
will be increased until either it reaches its maximum or the problem becomes bounded.
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